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<meta name="citation_issn" content="2210-271X" />
<meta name="citation_volume" content="1144" />
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<meta name="citation_publisher" content="Elsevier" />
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<meta name="citation_journal_title" content="Computational and Theoretical Chemistry" />
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<meta name="citation_doi" content="10.1016/j.comptc.2018.10.004" />
<meta name="dc.identifier" content="10.1016/j.comptc.2018.10.004" />
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<meta property="og:description" content="This work presents an implementation of the original orbital-free Hohenberg-Kohn density functional theory in a form that is able to predict chemical …" />
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<meta name="citation_title" content="Chemical bonding without orbitals" />
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<meta name="citation_publication_date" content="2018/11/15" />
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<title>Chemical bonding without orbitals - ScienceDirect</title>
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Contents" class="preview-sidebar u-show-from-lg col-lg-6"><div class="PreviewTableOfContents text-s"><h2 class="u-h4 preview-table-of-contents-title">Article preview</h2><ul class="preview-table-of-contents-list"><li id="preview-section-abstract-item" class><a class="anchor anchor-default" href="#preview-section-abstract"><span class="anchor-text">Abstract</span></a></li><li id="preview-section-introduction-item" class><a class="anchor anchor-default" href="#preview-section-introduction"><span class="anchor-text">Introduction</span></a></li><li id="preview-section-snippets-item" class><a class="anchor anchor-default" href="#preview-section-snippets"><span class="anchor-text">Section snippets</span></a></li><li id="preview-section-references-item" class><a class="anchor anchor-default" href="#preview-section-references"><span class="anchor-text">References (79)</span></a></li><li id="preview-section-cited-by-item" class><a class="anchor anchor-default" href="#preview-section-cited-by"><span class="anchor-text">Cited by (15)</span></a></li></ul></div></div><article class="col-lg-12 col-md-16 pad-left pad-right" lang="en"><div class="Publication" id="publication"><div class="publication-brand u-show-from-sm"><a class="anchor anchor-default" href="/journal/computational-and-theoretical-chemistry" title="Go to Computational and Theoretical Chemistry on ScienceDirect"><span class="anchor-text"><img class="publication-brand-image" src="https://sdfestaticassets-eu-west-1.sciencedirectassets.com/prod/5258de8cac0511407d267197e045479f0c04c8c0/image/elsevier-non-solus.png" alt="Elsevier" /></span></a></div><div class="publication-volume u-text-center"><h2 class="publication-title u-h3" id="publication-title"><a class="anchor publication-title-link anchor-navigation" href="/journal/computational-and-theoretical-chemistry" title="Go to Computational and Theoretical Chemistry on ScienceDirect"><span class="anchor-text">Computational and Theoretical Chemistry</span></a></h2><div class="text-xs"><a class="anchor anchor-default" href="/journal/computational-and-theoretical-chemistry/vol/1144/suppl/C" title="Go to table of contents for this volume/issue"><span class="anchor-text">Volume 1144</span></a>, <!-- -->15 November 2018<!-- -->, Pages 50-55</div></div><div class="publication-cover u-show-from-sm"><a class="anchor anchor-default" href="/journal/computational-and-theoretical-chemistry/vol/1144/suppl/C"><span class="anchor-text"><img class="publication-cover-image" src="https://ars.els-cdn.com/content/image/1-s2.0-S2210271X18X00206-cov150h.gif" alt="Computational and Theoretical Chemistry" /></span></a></div></div><div class="PageDivider"></div><h1 id="screen-reader-main-title" class="Head u-font-serif u-h2 u-margin-s-ver"><span class="title-text">Chemical bonding without orbitals</span></h1><div class="Banner" id="banner"><div class="wrapper truncated"><div class="AuthorGroups text-s"><div class="author-group" 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bonding in molecules. The method is completely parameter-free and does not require analytical functional approximations. Instead, the proposed method is based on the idea that atoms are meaningful pieces of a molecule and thus, a promolecule, build from frozen spherical atomic entities, serves as a suitable model for the latter. This idea is imposed on the physical equations, originating from density functional theory converted into a bifunctional formalism.</p><p id="sp020">The viewpoint proposed in this study offers a new strategic way of subsequent approximation levels in orbital-free density functional theory. In this work the zeroth order approximation is shown to predict chemical bonding in molecules, providing a concept of the chemical bond without involving orbitals.</p></div></div><div class="abstract graphical" id="ab005" lang="en"><h2 class="section-title u-h4 u-margin-l-top u-margin-xs-bottom">Graphical abstract</h2><div id="as005"><p id="sp005"><span class="display"><figure class="figure text-xs" id="f0030"><span><img src="https://ars.els-cdn.com/content/image/1-s2.0-S2210271X18305656-ga1.jpg" height="200" alt /><ol class="u-margin-s-bottom"><li><a class="anchor download-link u-font-sans u-display-inline anchor-default" href="https://ars.els-cdn.com/content/image/1-s2.0-S2210271X18305656-ga1_lrg.jpg" target="_blank" download title="Download high-res image (50KB)"><span class="anchor-text">Download : <span class="download-link-title">Download high-res image (50KB)</span></span></a></li><li><a class="anchor download-link u-font-sans u-display-inline anchor-default" href="https://ars.els-cdn.com/content/image/1-s2.0-S2210271X18305656-ga1.jpg" target="_blank" download title="Download full-size image"><span class="anchor-text">Download : <span class="download-link-title">Download full-size image</span></span></a></li></ol></span></figure></span></p></div></div></div></div><div id="preview-section-introduction"><div class="PageDivider"></div><div class="Introduction u-font-serif text-s u-margin-l-ver"><h2 class="u-h4 u-margin-s-bottom">Introduction</h2><section id="s0005"><p id="p0005">The chemical bond [1], [2], [3], [4] is a fundamental concept in chemistry and related sciences. Taking a very simplistic standpoint, the chemical bond somehow serves as glue, binding together the partaking atoms [5], [6]. One part of chemistry is to define and quantify chemical bonding in order to predict molecular stability and how the system will undergo possible changes induced by the environment. Unfortunately, the definition of that chemical bonding is ambiguous. This is due to the fact that atoms in molecules are no physical entities. From a puristic standpoint, the molecule is defined once the number of electrons and the nuclear positions are given. By solving the quantum mechanical equations [7], [5], the statistical distribution of the electrons in the field of the nuclei is determined. In that spirit, the molecule only consists as a whole, namely of all nuclei and all electrons, whereby the latter are usually (but not necessarily) described in terms of orbitals. The equations describe all orbitals at once and their solutions extend over the whole space. This form of wavefunction based quantum mechanics is said to be non-local [7], meaning that changes induced at one position in space might cause noticeable effects far away from the initial position. Such a non-local form of quantum mechanics is not suited for an atomic fragment approach, where those fragments are thought as being subjected to a physical interaction.</p><p id="p0010">Indeed a local from of quantum theory is needed, further allowing for a meaningful separation into atomic fragments. This local form of quantum theory is called density-functional theory (DFT) [8], [9]. In the Hohenberg-Kohn variant[10], the system is described by the electron density itself, rather than by orbitals. This theory is guided by a local causality principle [11], [12], [13], [14], [15], [16], [17], [18] (effective interactions decrease with increasing distance) and, as will be shown in this work, allows for a meaningful separation into atomic fragments, while keeping the initial physical equations unaltered. Although the original ideas of Hohenberg and Kohn dates back more than fifty years, a widespread use of orbital-free DFT has been hampered by the lack of sufficiently accurate functional approximations for the kinetic energy [19]. Whereas considerable progress has been made for one-dimensional systems [20], the treatment of molecular systems or solids requires appropriate three-dimensional functional approximations. First work was based on conventional gradient expansions [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], whereas recent functional development is mainly focused on techniques based on the generalized-gradient-approximation motivated by conjoint arguments [33], [34] or the fulfillment of additional constraints [35], [36], [19]. Information-theory motivated functionals [37], [38], [39], functionals based on response theory [40], [41], and expansions in terms of moment densities [42] have been studied. Recently, a very promising approach based on the Liu-Parr power series expansion [43] has been suggested by Ludeña and coworkers [44], [45], including a detailed investigation concerning the ability of their ansatz to represent the atomic shell structure. Despite much efforts, it still remains difficult to design appropriate functional approximations, which yield properly structured electron densities from variational procedure [46]. This failure can be attributed to an insufficient incorporation of the Pauli exclusion principle [27], [8], [19] in the Pauli kinetic energy and the corresponding Pauli potential [47]. Both terms were intensively studied in the literature [48], [49], [50], [51], [52], [53], [54], [55], [56] as they represent the unknown part of the kinetic energy functional and the corresponding potential, respectively. As a consequence, there is up to now no reliably working orbital-free density functional implementation for the waste variety of chemical substances. There are a few noticeable exceptions treating variational orbital-free density functional approximations [57], [58], [59], [60], [61]. However, widespread applications covering all types of chemical substances without need of recursive adjustment of technical details are still out of reach.</p><p id="p0015">This work offers a new strategy in orbital-free density functional theory based on the reformulation in terms of bifunctionals.</p></section></div></div><div id="preview-section-snippets"><div class="PageDivider"></div><div class="Snippets u-font-serif text-s"><h2 class="u-h4 u-margin-l-ver">Section snippets</h2><section><section id="s0010"><h2 class="section-title u-h4 u-margin-l-top u-margin-xs-bottom">Formulation of the problem</h2><p id="p0020">In 1964 Hohenberg and Kohn (HK) rigorously founded density functional theory (DFT) [10]. In their seminal paper they proved the one-to-one correspondence between the energy <em>E</em> of the system and the external potential due to the nuclei <span class="math"><math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">v</mi></mrow><mrow is="true"><mi is="true">Z</mi></mrow></msub></mrow></math></span> (first HK theorem) as well as the minimum principle of that energy functional <span class="math"><math><mrow is="true"><mi is="true">E</mi><mo stretchy="true" is="true">[</mo><mi is="true">ρ</mi><mo stretchy="true" is="true">]</mo></mrow></math></span> (second HK theorem) for all electron densities <span class="math"><math><mrow is="true"><mi is="true">ρ</mi></mrow></math></span> that are associated with some external potential. According to the first HK theorem, the electronic energy of a system <span class="math"><math><mrow is="true"><mi is="true">E</mi><mo stretchy="true" is="true">[</mo><mi is="true">ρ</mi><mo stretchy="true" is="true">]</mo></mrow></math></span>, given </p></section></section><section><section id="s0015"><h2 class="section-title u-h4 u-margin-l-top u-margin-xs-bottom">Reformulation in terms of bifunctionals</h2><p id="p0035">The conventional strategy in the design of energy functionals is to start with an ansatz for the energy in the form of an analytical expression that is an integral over a function <em>f</em>, which in some way contains the electron density (and possibly some other ingredients):<span class="display"><span id="e0020" class="formula"><span class="math"><math><mi is="true">F</mi><mo stretchy="true" is="true">[</mo><mi is="true">ρ</mi><mo stretchy="true" is="true">]</mo><mo is="true">=</mo><mo is="true">∫</mo><mi is="true">f</mi><mo stretchy="true" is="true">(</mo><mi is="true">ρ</mi><mo stretchy="true" is="true">(</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">r</mi></mrow><mrow is="true"><mo stretchy="true" is="true">→</mo></mrow></mover><mo stretchy="true" is="true">)</mo><mo is="true">,</mo><mo is="true">…</mo><mo stretchy="true" is="true">)</mo><mspace width="0.12em" is="true"></mspace><mtext is="true">d</mtext><mover accent="true" is="true"><mrow is="true"><mi is="true">r</mi></mrow><mrow is="true"><mo stretchy="true" is="true">→</mo></mrow></mover><mspace width="0.25em" is="true"></mspace><mspace width="0.25em" is="true"></mspace><mtext is="true">.</mtext></math></span></span></span>From that functional, the potential, being the functional derivative, is obtained analytically, cf. Eq. (3). As mentioned, no ansatz for the kinetic energy has been found that is able to predict chemical</p></section></section><section><section id="s0020"><h2 class="section-title u-h4 u-margin-l-top u-margin-xs-bottom">The chemical bond in the atomic fragment approximation</h2><p id="p0055">From a chemist’s viewpoint it seems natural to interpret [71], [72], [73] large aggregates as being composed from atomic fragments [74], [75], [76], [77] that somehow hold together. In the following it is shown how to link such a viewpoint with the quantum mechanical equations based on the bifunctional formalism and the atomic fragment approach.</p><p id="p0060">A so-called promolecule [78], [76] is the sum of (usually spherical) atoms centered at the positions of the nuclei for the actual molecule of interest.</p></section></section><section><section id="s0025"><h2 class="section-title u-h4 u-margin-l-top u-margin-xs-bottom">Discussion</h2><p id="p0095">This work presents an orbital-free implementation of the original Hohenberg-Kohn density functional theory, that is able to predict bonding in molecules. The proposed method is completely parameter-free and does not require analytical ansatzes for the energy functionals. Instead, it is based on the idea that atoms are meaningful pieces of a molecule and thus, a promolecule serves as a good model for the latter. This idea is imposed on the physical equations, originating from density functional</p></section></section><section><section id="ak005"><h2 id="st055" class="u-h4 u-margin-l-top u-margin-xs-bottom">Acknowledgments</h2><p id="p0155">The author wishes to thank Dr. M. Kohout for fruitful discussions and substantial encouragement over years. Prof. Dr. M. Ruck is greatly acknowledged for academic support and valuable hints on the manuscript. The <span id="gp005">Technische Universität Dresden</span> is acknowledged for funding in terms of a Maria-Reiche fellowship.</p></section></section></div></div><div class="related-content-links u-hide-from-md"><button type="button" class="button button-anchor" aria-disabled="false"><span class="button-text">Recommended articles</span></button></div><div class="Tail text-s"></div><div id="preview-section-references"><div class="paginatedReferences u-font-serif text-s"><div class="PageDivider"></div><header><h2 class="u-h4 u-margin-l-ver"><span>References</span><span> (79)</span></h2></header><ul><li class="bib-reference u-margin-s-bottom"><span class="u-font-sans"><span class="author u-font-sans"><span>V. </span>Karasiev</span><em> et al.</em></span><h3><a class="anchor title anchor-default" href="/science/article/pii/S0065327615000052" target="_self"><span class="anchor-text">Frank discussion o the status of ground-state orbital-free DFT</span></a></h3><span class="host u-clr-grey6 u-font-sans"><div class="series"><h3 class="title">Adv. 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Phys.</h3></div><div class="series">(2009)</div></span></li><li class="bib-reference u-margin-s-bottom"><span class="u-font-sans"><span class="author u-font-sans"><span>V. </span>Karasiev</span><em> et al.</em></span><h3 class="title">Progress on new approaches to old ideas: orbital-free density functionals</h3><span class="host u-clr-grey6 u-font-sans"></span></li></div><button class="button-alternative button-alternative-secondary text-m u-font-sans u-margin-l-bottom button-alternative-icon-left" type="button" id="show-more-refs-btn"><span class="button-alternative-icon"><svg focusable="false" viewBox="0 0 92 128" width="17.25" height="24" class="icon icon-navigate-down u-fill-grey8"><path d="m1 51l7-7 38 38 38-38 7 7-45 45z"></path></svg></span><span class="button-alternative-text">View more references</span></button></div></div><div id="preview-section-cited-by"><section aria-label="Cited by" class="ListArticles preview"><div class="PageDivider"></div><header id="citing-articles-header"><h2 class="u-h4 u-margin-l-ver u-font-serif">Cited by (15)</h2></header><div aria-describedby="citing-articles-header"><div class="citing-articles u-margin-l-bottom"><ul><li class="ListArticleItem u-margin-l-bottom"><div class="sub-heading u-margin-xs-bottom"><h3 class="u-font-serif" id="citing-articles-article-0-title"><a class="anchor anchor-default" href="/science/article/pii/S2210271X19300842"><span class="anchor-text">A study of the basis set dependence of the bifunctional expression of the non-interacting kinetic energy for atomic systems</span></a></h3><div class="text-s">2019, Computational and Theoretical Chemistry</div><div class="CitedSection u-margin-s-top"><div class="u-margin-s-left"><div class="cite-header text-s u-text-italic u-font-sans">Citation Excerpt :</div><p class="u-font-serif text-xs">This approach requires appropriate density-based approximations for the kinetic energy. Recently, a parameter-free approximation based on a bifunctional expression was introduced [8,9]. In contrast to other known approaches in the literature targeting the energy functional directly [10–41], the proposed bifunctional approach is based on an approximation of the first functional derivative.</p></div></div></div><div class="buttons text-s"><button class="button-link button-link-secondary button-link-icon-right" type="button" data-aa-button="sd:product:journal:article:location=citing-articles:type=view-details" aria-describedby="citing-articles-article-0-title" aria-controls="citing-articles-article-0" aria-expanded="false"><span class="button-link-text">Show abstract</span><svg focusable="false" viewBox="0 0 92 128" width="17.25" height="24" class="icon icon-navigate-down"><path d="m1 51l7-7 38 38 38-38 7 7-45 45z"></path></svg></button></div><div class="u-display-none" aria-hidden="true"><div class="abstract u-margin-xs-top u-margin-m-bottom u-font-serif text-s" id="reference-abstract"><div class="u-margin-ver-m"><p id="sp015">The non-interacting kinetic energy is divided into the von Weizsäcker term and the Pauli kinetic energy. Whereas the von Weizsäcker energy is known in terms of the density, the Pauli kinetic energy is not. Consequently, its functional derivative can only be determined formally, at the solution point, from a given set of Kohn-Sham eigenfunctions. Since in a practical calculation the solution point is never reached exactly, the formal functional derivative is evaluated only in proximity of the solution point. Therefore, bifunctional expressions involving the corresponding potential are approximate. In this study the atoms from H - Xe are examined, showing that the energy deviation between the bifunctional expression and the orbital-based kinetic energy density is of a few hundred millihartrees for a quadruple basis set, while it can reach the order of a few hartrees when employing basis sets of less quality.</p></div></div></div></li><li class="ListArticleItem u-margin-l-bottom"><div class="sub-heading u-margin-xs-bottom"><h3 class="u-font-serif" id="citing-articles-article-1-title"><a class="anchor anchor-default" href="https://doi.org/10.1107/S2052520621005540" target="_blank"><span class="anchor-text">Developing orbital-free quantum crystallography: The local potentials and associated partial charge densities</span><svg focusable="false" viewBox="0 0 78 128" aria-label="Opens in new window" width="1em" height="1em" class="icon icon-arrow-up-right arrow-external-link"><path d="m4 36h57.07l-59.5 59.5 7.07 7.08 59.36-59.36v56.78h1e1v-74h-74z"></path></svg></a></h3><div 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Recently, a parameter-free approximation based on a bifunctional expression was introduced [8,9]. In contrast to other known approaches in the literature targeting the energy functional directly [10–41], the proposed bifunctional approach is based on an approximation of the first functional derivative."],"thirdParty":false,"volume":"Volume 1155","abstract":{"$$":[{"$$":[{"$$":[{"#name":"attachment-eid","_":"1-s2.0-S2210271X19300842-ga1.sml"},{"#name":"ucs-locator","_":"https://s3.amazonaws.com/prod-ucs-content-store-us-east/content/pii:S2210271X19300842/ga1/THUMBNAIL/image/gif/a27391a194e40a949f39246b21f473d4/ga1.sml"},{"#name":"file-basename","_":"ga1"},{"#name":"abstract-attachment","_":"true"},{"#name":"filename","_":"ga1.sml"},{"#name":"extension","_":"sml"},{"#name":"filesize","_":"6304"},{"#name":"pixel-height","_":"149"},{"#name":"pixel-width","_":"219"},{"#name":"attachment-type","_":"IMAGE-THUMBNAIL"}],"$":{"xmlns:xocs":true},"#name":"attachment"},{"$$":[{"#name":"attachment-eid","_":"1-s2.0-S2210271X19300842-ga1.jpg"},{"#name":"ucs-locator","_":"https://s3.amazonaws.com/prod-ucs-content-store-us-east/content/pii:S2210271X19300842/ga1/DOWNSAMPLED/image/jpeg/686597a7871188382549a56e6d9de8cf/ga1.jpg"},{"#name":"file-basename","_":"ga1"},{"#name":"abstract-attachment","_":"true"},{"#name":"filename","_":"ga1.jpg"},{"#name":"extension","_":"jpg"},{"#name":"filesize","_":"13252"},{"#name":"pixel-height","_":"200"},{"#name":"pixel-width","_":"293"},{"#name":"attachment-type","_":"IMAGE-DOWNSAMPLED"}],"$":{"xmlns:xocs":true},"#name":"attachment"},{"$$":[{"#name":"attachment-eid","_":"1-s2.0-S2210271X19300842-ga1_lrg.jpg"},{"#name":"ucs-locator","_":"https://s3.amazonaws.com/prod-ucs-content-store-us-east/content/pii:S2210271X19300842/ga1/HIGHRES/image/jpeg/f109e8e5cc6695e0feae159da06988c9/ga1_lrg.jpg"},{"#name":"file-basename","_":"ga1"},{"#name":"abstract-attachment","_":"true"},{"#name":"filename","_":"ga1_lrg.jpg"},{"#name":"extension","_":"jpg"},{"#name":"filesize","_":"83430"},{"#name":"pixel-height","_":"886"},{"#name":"pixel-width","_":"1300"},{"#name":"attachment-type","_":"IMAGE-HIGH-RES"}],"$":{"xmlns:xocs":true},"#name":"attachment"}],"#name":"attachments"},{"$$":[{"$":{"id":"st030"},"#name":"section-title","_":"Graphical abstract"},{"$$":[{"$$":[{"$$":[{"$$":[{"$":{"role":"http://data.elsevier.com/vocabulary/ElsevierContentTypes/23.4","xmlns:xlink":true,"id":"lk005","href":"pii:S2210271X19300842/ga1","locator":"ga1"},"#name":"link"}],"$":{"id":"f0025"},"#name":"figure"}],"#name":"display"}],"$":{"view":"all","id":"sp005"},"#name":"simple-para"}],"$":{"view":"all","id":"as005"},"#name":"abstract-sec"}],"$":{"view":"all","id":"ab005","lang":"en","class":"graphical"},"#name":"abstract"},{"$$":[{"$":{"id":"st035"},"#name":"section-title","_":"Highlights"},{"$$":[{"$$":[{"$$":[{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0100"},"#name":"para","_":"Basis set dependence for non-analytical kinetic energy functionals are investigated."}],"$":{"id":"u0005"},"#name":"list-item"},{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0105"},"#name":"para","_":"Only the Pauli kinetic energy is affected by basis set effects."}],"$":{"id":"u0010"},"#name":"list-item"},{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0110"},"#name":"para","_":"Non-analytical functional expressions for the Weizsäcker kinetic energy are exact."}],"$":{"id":"u0015"},"#name":"list-item"}],"$":{"id":"l0005"},"#name":"list"}],"$":{"view":"all","id":"sp010"},"#name":"simple-para"}],"$":{"view":"all","id":"as010"},"#name":"abstract-sec"}],"$":{"view":"all","id":"ab010","lang":"en","class":"author-highlights"},"#name":"abstract"},{"$$":[{"$":{"id":"st040"},"#name":"section-title","_":"Abstract"},{"$$":[{"$":{"view":"all","id":"sp015"},"#name":"simple-para","_":"The non-interacting kinetic energy is divided into the von Weizsäcker term and the Pauli kinetic energy. Whereas the von Weizsäcker energy is known in terms of the density, the Pauli kinetic energy is not. Consequently, its functional derivative can only be determined formally, at the solution point, from a given set of Kohn-Sham eigenfunctions. Since in a practical calculation the solution point is never reached exactly, the formal functional derivative is evaluated only in proximity of the solution point. Therefore, bifunctional expressions involving the corresponding potential are approximate. In this study the atoms from H - Xe are examined, showing that the energy deviation between the bifunctional expression and the orbital-based kinetic energy density is of a few hundred millihartrees for a quadruple basis set, while it can reach the order of a few hartrees when employing basis sets of less quality."}],"$":{"view":"all","id":"as015"},"#name":"abstract-sec"}],"$":{"view":"all","id":"ab015","lang":"en","class":"author"},"#name":"abstract"}],"$":{"xmlns:ce":true,"xmlns:dm":true,"xmlns:sb":true},"#name":"abstracts"},"pdf":{"urlType":"download","url":"/science/article/pii/S2210271X19300842/pdfft?md5=b6d7b4c354b6283d8529a519905e9cb2&pid=1-s2.0-S2210271X19300842-main.pdf"}},{"articleTitle":"Developing orbital-free quantum crystallography: The local potentials and associated partial charge densities","authors":"Tsirelson V., Stash A.","doi":"10.1107/S2052520621005540","externalArticle":true,"openAccess":0,"page":"467-477","pii":"S2052520621005540","publicationTitle":"Acta 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chemical bond [1], [2], [3], [4] is a fundamental concept in chemistry and related sciences. Taking a very simplistic standpoint, the chemical bond somehow serves as glue, binding together the partaking atoms [5], [6]. One part of chemistry is to define and quantify chemical bonding in order to predict molecular stability and how the system will undergo possible changes induced by the environment. Unfortunately, the definition of that chemical bonding is ambiguous. This is due to the fact that atoms in molecules are no physical entities. From a puristic standpoint, the molecule is defined once the number of electrons and the nuclear positions are given. By solving the quantum mechanical equations [7], [5], the statistical distribution of the electrons in the field of the nuclei is determined. In that spirit, the molecule only consists as a whole, namely of all nuclei and all electrons, whereby the latter are usually (but not necessarily) described in terms of orbitals. The equations describe all orbitals at once and their solutions extend over the whole space. This form of wavefunction based quantum mechanics is said to be non-local [7], meaning that changes induced at one position in space might cause noticeable effects far away from the initial position. Such a non-local form of quantum mechanics is not suited for an atomic fragment approach, where those fragments are thought as being subjected to a physical interaction."},{"#name":"para","$":{"id":"p0010","view":"all"},"_":"Indeed a local from of quantum theory is needed, further allowing for a meaningful separation into atomic fragments. This local form of quantum theory is called density-functional theory (DFT) [8], [9]. In the Hohenberg-Kohn variant[10], the system is described by the electron density itself, rather than by orbitals. This theory is guided by a local causality principle [11], [12], [13], [14], [15], [16], [17], [18] (effective interactions decrease with increasing distance) and, as will be shown in this work, allows for a meaningful separation into atomic fragments, while keeping the initial physical equations unaltered. Although the original ideas of Hohenberg and Kohn dates back more than fifty years, a widespread use of orbital-free DFT has been hampered by the lack of sufficiently accurate functional approximations for the kinetic energy [19]. Whereas considerable progress has been made for one-dimensional systems [20], the treatment of molecular systems or solids requires appropriate three-dimensional functional approximations. First work was based on conventional gradient expansions [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], whereas recent functional development is mainly focused on techniques based on the generalized-gradient-approximation motivated by conjoint arguments [33], [34] or the fulfillment of additional constraints [35], [36], [19]. Information-theory motivated functionals [37], [38], [39], functionals based on response theory [40], [41], and expansions in terms of moment densities [42] have been studied. Recently, a very promising approach based on the Liu-Parr power series expansion [43] has been suggested by Ludeña and coworkers [44], [45], including a detailed investigation concerning the ability of their ansatz to represent the atomic shell structure. Despite much efforts, it still remains difficult to design appropriate functional approximations, which yield properly structured electron densities from variational procedure [46]. This failure can be attributed to an insufficient incorporation of the Pauli exclusion principle [27], [8], [19] in the Pauli kinetic energy and the corresponding Pauli potential [47]. Both terms were intensively studied in the literature [48], [49], [50], [51], [52], [53], [54], [55], [56] as they represent the unknown part of the kinetic energy functional and the corresponding potential, respectively. As a consequence, there is up to now no reliably working orbital-free density functional implementation for the waste variety of chemical substances. There are a few noticeable exceptions treating variational orbital-free density functional approximations [57], [58], [59], [60], [61]. However, widespread applications covering all types of chemical substances without need of recursive adjustment of technical details are still out of reach."},{"#name":"para","$":{"id":"p0015","view":"all"},"_":"This work offers a new strategy in orbital-free density functional theory based on the reformulation in terms of bifunctionals."}]}]},{"#name":"section","$$":[{"#name":"section","$":{"xmlns:ce":true,"xmlns:mml":true,"xmlns:xs":true,"xmlns:xlink":true,"xmlns:xocs":true,"xmlns:tb":true,"xmlns:xsi":true,"xmlns:cals":true,"xmlns:sb":true,"xmlns:sa":true,"xmlns:ja":true,"xmlns":true,"id":"s0010","view":"all"},"$$":[{"#name":"label","_":"2"},{"#name":"section-title","$":{"id":"st010"},"_":"Formulation of the problem"},{"#name":"para","$":{"id":"p0020","view":"all"},"$$":[{"#name":"__text__","_":"In 1964 Hohenberg and Kohn (HK) rigorously founded density functional theory (DFT) [10]. 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Eq. (3). As mentioned, no ansatz for the kinetic energy has been found that is able to predict chemical"}]}]}]},{"#name":"section","$$":[{"#name":"section","$":{"xmlns:ce":true,"xmlns:mml":true,"xmlns:xs":true,"xmlns:xlink":true,"xmlns:xocs":true,"xmlns:tb":true,"xmlns:xsi":true,"xmlns:cals":true,"xmlns:sb":true,"xmlns:sa":true,"xmlns:ja":true,"xmlns":true,"id":"s0020","view":"all"},"$$":[{"#name":"label","_":"4"},{"#name":"section-title","$":{"id":"st020"},"_":"The chemical bond in the atomic fragment approximation"},{"#name":"para","$":{"id":"p0055","view":"all"},"_":"From a chemist’s viewpoint it seems natural to interpret [71], [72], [73] large aggregates as being composed from atomic fragments [74], [75], [76], [77] that somehow hold together. In the following it is shown how to link such a viewpoint with the quantum mechanical equations based on the bifunctional formalism and the atomic fragment approach."},{"#name":"para","$":{"id":"p0060","view":"all"},"_":"A so-called promolecule [78], [76] is the sum of (usually spherical) atoms centered at the positions of the nuclei for the actual molecule of interest."}]}]},{"#name":"section","$$":[{"#name":"section","$":{"xmlns:ce":true,"xmlns:mml":true,"xmlns:xs":true,"xmlns:xlink":true,"xmlns:xocs":true,"xmlns:tb":true,"xmlns:xsi":true,"xmlns:cals":true,"xmlns:sb":true,"xmlns:sa":true,"xmlns:ja":true,"xmlns":true,"id":"s0025","view":"all"},"$$":[{"#name":"label","_":"5"},{"#name":"section-title","$":{"id":"st025"},"_":"Discussion"},{"#name":"para","$":{"id":"p0095","view":"all"},"_":"This work presents an orbital-free implementation of the original Hohenberg-Kohn density functional theory, that is able to predict bonding in molecules. The proposed method is completely parameter-free and does not require analytical ansatzes for the energy functionals. Instead, it is based on the idea that atoms are meaningful pieces of a molecule and thus, a promolecule serves as a good model for the latter. This idea is imposed on the physical equations, originating from density functional"}]}]},{"#name":"section","$$":[{"#name":"acknowledgment","$":{"xmlns:ce":true,"xmlns:mml":true,"xmlns:xs":true,"xmlns:xlink":true,"xmlns:xocs":true,"xmlns:tb":true,"xmlns:xsi":true,"xmlns:cals":true,"xmlns:sb":true,"xmlns:sa":true,"xmlns:ja":true,"xmlns":true,"id":"ak005","view":"all"},"$$":[{"#name":"section-title","$":{"id":"st055"},"_":"Acknowledgments"},{"#name":"para","$":{"id":"p0155","view":"all"},"$$":[{"#name":"__text__","_":"The author wishes to thank Dr. M. Kohout for fruitful discussions and substantial encouragement over years. Prof. Dr. M. Ruck is greatly acknowledged for academic support and valuable hints on the manuscript. The "},{"#name":"grant-sponsor","$":{"id":"gp005","type":"simple","role":"http://www.elsevier.com/xml/linking-roles/grant-sponsor"},"_":"Technische Universität Dresden"},{"#name":"__text__","_":" is acknowledged for funding in terms of a Maria-Reiche fellowship."}]}]}]}],"floats":[],"footnotes":[],"attachments":[]},"questionsAndAnswers":{},"rawtext":"","recommendations":{"articles":[{"pii":"S1386947718312542","doi":"10.1016/j.physe.2018.10.022","journalTitle":"Physica E: Low-dimensional Systems and Nanostructures","publicationYear":"2019","publicationDate":"2019-02-01","volumeSupText":"Volume 106","articleNumber":"","pageRange":"1-4","trace-token":"AAAAQPOAhzHPzmDEyWKxztRR1mTsAwQeqz4-H79FZRWVcLiBro07aOQDS4ve8reAnbnxz0PsGrwD3__IPunOy2KQvs2KxfwbdqGHK_MiTwUqtu_rnWztcA","authors":{"content":[{"#name":"author-group","$":{"id":"augrp0010"},"$$":[{"#name":"author","$":{"id":"au1","author-id":"S1386947718312542-2a967295bbb73415523fd30e83e1955f"},"$$":[{"#name":"given-name","_":"A.L."},{"#name":"surname","_":"Vartanian"},{"#name":"e-address","$":{"xmlns:xlink":true,"type":"email","href":"mailto:vardan@ysu.am","id":"eadd0010"},"_":"vardan@ysu.am"}]},{"#name":"author","$":{"id":"au2","author-id":"S1386947718312542-2015d1c3552b916e68aec575b19bbcc6"},"$$":[{"#name":"given-name","_":"A.L."},{"#name":"surname","_":"Asatryan"},{"#name":"e-address","$":{"xmlns:xlink":true,"type":"email","href":"mailto:annaa@ysu.am","id":"eadd0015"},"_":"annaa@ysu.am"}]},{"#name":"author","$":{"id":"au3","author-id":"S1386947718312542-95c350c6bf20ab1d3950551f76182ce1"},"$$":[{"#name":"given-name","_":"A.A."},{"#name":"surname","_":"Kirakosyan"},{"#name":"e-address","$":{"xmlns:xlink":true,"type":"email","href":"mailto:kirakosyan@ysu.am","id":"eadd0020"},"_":"kirakosyan@ysu.am"}]},{"#name":"author","$":{"id":"au4","author-id":"S1386947718312542-086ec316b85a3a3ee121f164e67b77ae"},"$$":[{"#name":"given-name","_":"K.A."},{"#name":"surname","_":"Vardanyan"},{"#name":"cross-ref","$":{"refid":"cor1","id":"crosref0010"},"$$":[{"#name":"sup","$":{"loc":"post"},"_":"∗"}]},{"#name":"e-address","$":{"xmlns:xlink":true,"type":"email","href":"mailto:karvard@ysu.am","id":"eadd0025"},"_":"karvard@ysu.am"}]},{"#name":"affiliation","$":{"id":"aff1","affiliation-id":"S1386947718312542-a1fabc2e74484983d9e73dab0d7abab0"},"$$":[{"#name":"textfn","_":"Department of Solid State Physics, Yerevan State University, 1, Al. Manoogian, Yerevan, 0025, Armenia"},{"#name":"affiliation","$":{"xmlns:sa":true},"$$":[{"#name":"organization","_":"Department of Solid State Physics"},{"#name":"organization","_":"Yerevan State University"},{"#name":"address-line","_":"1, Al. 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We calculate the binding energies of the ground and excited donor impurity states in the effective-mass approximation using the variational method for various values of the quantum well width, electric field strength, and donor positions within the well. Theoretical results indicate that the role of image charges is especially important for small widths of a quantum well. In contrast to the ground state, the influence of image charges on the binding energy of excited states is significant only at small values of the electric field. Therefore such fields can serve as a means to control the transition energies between ground and first excited states."}],"$":{"view":"all","id":"abssec0010"},"#name":"abstract-sec"}],"$":{"view":"all","id":"abs0010","lang":"en","class":"author"},"#name":"abstract"},{"$$":[{"$":{"id":"sectitle0015"},"#name":"section-title","_":"Highlights"},{"$$":[{"$$":[{"$$":[{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0010"},"#name":"para","_":"Effect of image charges on the spectrum of a donor in quantum well is studied."}],"$":{"id":"u0010"},"#name":"list-item"},{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0015"},"#name":"para","_":"The image charges are especially important for small widths of a quantum well."}],"$":{"id":"u0015"},"#name":"list-item"},{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0020"},"#name":"para","_":"Binding energies decreases sharply for positions closer to the boundary with 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efficient computation of Coulomb interactions in charged particle systems is of great importance in the field of molecular dynamics simulations. It is widely known that an approximation can be realized based on the Ewald summation approach and the fast Fourier transform (FFT). In the present paper we consider particle systems containing a mixture of "},{"#name":"italic","_":"N"},{"#name":"__text__","_":" point charges as well as point dipoles. New cutoff errors in the Ewald summation formulas concerning charge–dipole interactions are derived and, moreover, validated by numerical examples. Furthermore, we present for the first time an "},{"$$":[{"$":{"mathvariant":"script"},"#name":"mi","_":"O"},{"$":{"stretchy":"false"},"#name":"mo","_":"("},{"#name":"mi","_":"N"},{"$":{"mathvariant":"normal"},"#name":"mi","_":"log"},{"#name":"mo","_":"⁡"},{"#name":"mi","_":"N"},{"$":{"stretchy":"false"},"#name":"mo","_":")"}],"$":{"overflow":"scroll","xmlns:mml":true,"altimg":"si1.gif"},"#name":"math"},{"#name":"__text__","_":" particle mesh algorithm for computing mixed charge–dipole interactions based on the FFT for nonequispaced data (NFFT). We present first numerical results for charge–dipole systems, showing that the introduced method can be tuned to a high precision and verifying the "},{"$$":[{"$":{"mathvariant":"script"},"#name":"mi","_":"O"},{"$":{"stretchy":"false"},"#name":"mo","_":"("},{"#name":"mi","_":"N"},{"$":{"mathvariant":"normal"},"#name":"mi","_":"log"},{"#name":"mo","_":"⁡"},{"#name":"mi","_":"N"},{"$":{"stretchy":"false"},"#name":"mo","_":")"}],"$":{"overflow":"scroll","xmlns:mml":true,"altimg":"si1.gif"},"#name":"math"},{"#name":"__text__","_":" scaling. In order to calculate the interactions with dipoles efficiently, two new variants of the NFFT, namely the Hessian NFFT as well as the adjoint gradient NFFT, are derived and implemented. In the context of NFFT, these new variants are of great importance on their own. The presented particle mesh method is an extension of the particle–particle NFFT (P"},{"$":{"loc":"post"},"#name":"sup","_":"2"},{"#name":"__text__","_":"NFFT) framework. 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The modification was modeled by the variations of the NCoN and ClCoCl angles, by the rotation of the chlorido ligands with respect to other ligand groups as well as by the strength of the "},{"#name":"italic","_":"N"},{"#name":"__text__","_":"-donor ligands. 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The advancements have opened up many possibilities in applications of sensors, transformers, and microwave devices. We presented here a new quick test of ME composites using nomographs and showed its use in applications where an approximate answer is appropriate and useful. To draw the graphs for ME voltage coefficients, we derived approximate expressions in explicit form for magnetically induced ME effect for different operational modes and laminate composite configurations including symmetrical and asymmetrical structures."}],"$":{"view":"all","id":"abs0010"},"#name":"abstract-sec"}],"$":{"view":"all","id":"ab0010","class":"author"},"#name":"abstract"},{"$$":[{"$":{"id":"sect0010"},"#name":"section-title","_":"Highlights"},{"$$":[{"$$":[{"$$":[{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0005"},"#name":"para","_":"A new quick test of magnetoelectric composites using nomographs is proposed."}],"$":{"id":"u0005"},"#name":"list-item"},{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0010"},"#name":"para","_":"Approximate expressions in explicit form for magnetically induced magnetoelectric effect for different operational modes are derived."}],"$":{"id":"u0010"},"#name":"list-item"},{"$$":[{"#name":"label","_":"•"},{"$":{"view":"all","id":"p0015"},"#name":"para","_":"Using the nomographs enables one to predict the volume fractions of composite components and sample geometry for specific composition to provide the strongest magnetoelectric coupling."}],"$":{"id":"u0015"},"#name":"list-item"}],"$":{"id":"li0005"},"#name":"list"}],"$":{"view":"all","id":"sp0080"},"#name":"simple-para"}],"$":{"view":"all","id":"abs0015"},"#name":"abstract-sec"}],"$":{"view":"all","id":"ab0015","class":"author-highlights"},"#name":"abstract"}],"$":{"xmlns:ce":true,"xmlns:dm":true,"xmlns:sb":true},"#name":"abstracts"},"pdf":{"urlType":null},"iss-first":"","vol-first":"412","isThirdParty":false,"language":"en","issn-primary-unformatted":"03048853","issn-primary-formatted":"0304-8853"},{"pii":"B9780124095472110273","doi":"10.1016/B978-0-12-409547-2.11027-3","journalTitle":"Encyclopedia of Analytical Science","publicationYear":"2019","publicationDate":"2019-01-01","volumeSupText":"","articleNumber":"","pageRange":"270-280","trace-token":"AAAAQPOAhzHPzmDEyWKxztRR1mTsBLgjFj0XMN8PeA6H1bQV7xXxce-a3WqFcaFxc87IvIXlqxTPJbRZIdWiCLZMKof__jZud7PeOZ1KCx1jU5tf7eI6Nw","authors":{"content":[{"#name":"author-group","$":{"id":"ag0010"},"$$":[{"#name":"author","$":{"id":"au0010","author-id":"B9780124095472110273-cecf9fa3bffe387a44f849e13e127e2c"},"$$":[{"#name":"given-name","_":"Jacqui L."},{"#name":"surname","_":"Adcock"},{"#name":"e-address","$":{"xmlns:xlink":true,"href":"mailto:jacqui.adcock@deakin.edu.au","type":"email","id":"em0010"},"_":"jacqui.adcock@deakin.edu.au"}]},{"#name":"author","$":{"id":"au0015","author-id":"B9780124095472110273-855a387ffd6521dbac51674e205330f6"},"$$":[{"#name":"given-name","_":"Neil W."},{"#name":"surname","_":"Barnett"},{"#name":"e-address","$":{"xmlns:xlink":true,"href":"mailto:barnie@deakin.edu.au","type":"email","id":"em0015"},"_":"barnie@deakin.edu.au"}]},{"#name":"author","$":{"id":"au0020","author-id":"B9780124095472110273-289964e189585dcb1fdcfceef1712ea4"},"$$":[{"#name":"given-name","_":"Paul S."},{"#name":"surname","_":"Francis"},{"#name":"e-address","$":{"xmlns:xlink":true,"href":"mailto:paul.francis@deakin.edu.au","type":"email","id":"em0020"},"_":"paul.francis@deakin.edu.au"}]},{"#name":"affiliation","$":{"id":"af0010","affiliation-id":"B9780124095472110273-05628f945890871118df150a0f14179a"},"$$":[{"#name":"textfn","$":{"id":"tn0010"},"_":"Deakin University, Geelong, VIC, Australia"},{"#name":"affiliation","$":{"xmlns:sa":true},"$$":[{"#name":"organization","_":"Deakin University"},{"#name":"city","_":"Geelong"},{"#name":"state","_":"VIC"},{"#name":"country","_":"Australia"}]}]},{"#name":"footnote","$":{"id":"fn0010"},"$$":[{"#name":"label","_":"☆"},{"#name":"note-para","$":{"id":"np0010","view":"all"},"$$":[{"#name":"italic","_":"Change History"},{"#name":"__text__","_":": September 2018. Neil W Barnett expanded the Historical Perspectives section. Jacqui L Adcock, Neil W Barnett and Paul S Francis revised numerous aspects of the text and updated and added references. January 2014. Neil W Barnett added an abstract. Jacqui L Adcock and Paul S Francis added references and updates throughout text."}]},{"#name":"note-para","$":{"id":"notep0010","view":"all"},"_":"This is an update of Jacqui L. Adcock, Neil W. Barnett, Paul S. Francis, Luminescence: Overview, Reference Module in Chemistry, Molecular Sciences and Chemical Engineering, Elsevier, 2014."}]}]}],"floats":[],"footnotes":[{"#name":"footnote","$":{"id":"fn0010"},"$$":[{"#name":"label","_":"☆"},{"#name":"note-para","$":{"id":"np0010","view":"all"},"$$":[{"#name":"italic","_":"Change History"},{"#name":"__text__","_":": September 2018. Neil W Barnett expanded the Historical Perspectives section. Jacqui L Adcock, Neil W Barnett and Paul S Francis revised numerous aspects of the text and updated and added references. 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Contributions from noteworthy scientific pioneers are also summarized. The principles of molecular photoluminescence are presented with reference to excited state multiplicity and lifetimes, internal conversion, intersystem crossing, fluorescence, phosphorescence and quantum yield. Structural and environmental influences on photoluminescence are succinctly examined together with the relationship between emission intensity and analyte concentration. Monitoring of corrected excitation and emission spectra is discussed with reference to Stokes shift and Rayleigh scattering. 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